The springs have it...
It would be nice if you didn't have to pull the tranny and replace the clutch.
But you do.
The clutch springs (or something else, the brackets that hold the spring) has broken and is sometimes binding up (preventing the clutch from not spinning) which will not allow the transmission to stop spinning. While it's spinning, you can't get it out of gear nor into another gear. This is the key symptom. When you stop the engine, you can move the gears around (because the tranny is not spinning). Start the car, no more shifty.
When you press the clutch, it is activating the master, then slave cylinders which is pressing an arm and then on the throw-out bearing which is releasing pressure on the clutch plate.
This SHOULD allow the clutch to stop spinning. However, debris is jammed between the clutch and the spinning flywheel. The clutch is doing its job. The throw-out bearing is being pressed and the the fingers are moving.
BUT, the junk in there is binding up, and sometimes freeing up (when you start the car).
A number of us have had this experience in the past with one car or another. (My experience, with symptoms exactly like yours, was with a 1971 Celica...)
The spring may scratch the pressure plate such that you need to machine it, and or replace it.
Not an impossible job, just a fairly time-consuming job that requires some lifting and swearing.
Good luck.
Chessie